Turning free variables into arguments.
As an example, consider the following worker wrapper function, which computes the truncated square root of an integer:
isqrt :: Integer -> Integer isqrt n | n < 0 = error "isqrt" | otherwise = isqrt' ((n+1) `div` 2) where isqrt' s | s*s <= n && n < (s+1)*(s+1) = s | otherwise = isqrt' ((s + (n `div` s)) `div` 2)
Suppose that you find that the worker function,
isqrt', might be useful somewhere else (say, in a context where a better initial estimate is known). You would like to use the lifting pattern to raise it to the top level. However,
isqrt' contains a free variable,
n, which is bound in the outer function. What to do?
The solution is to promote
n to be an argument of
isqrt'. The refactored code might look like this:
isqrt :: Integer -> Integer isqrt n | n < 0 = error "isqrt" | otherwise = isqrt' n ((n+1) `div` 2) where isqrt' n s | s*s <= n && n < (s+1)*(s+1) = s | otherwise = isqrt' n ((s + (n `div` s)) `div` 2)
The isqrt' function may now be safely lifted to the top-level.
Naive lambda lifting can cause a program to be less lazy. Consider, for example:
f x y = g x + g (2*x) where g x = sqrt y + x
If you want to lift the definition of
g, you might be tempted to write:
f x y = g y x + g y (2*x) where g y x = sqrt y + x
However, this would mean that
sqrt y is evaluated twice, whereas in the first program, it would be evaluated only once. A more efficient approach is not to lift out the variable
y, but rather the expression
f x y = let sy = sqrt y in g sy x + g sy (2*x) where g sy x = sy + x
An expression of this sort which only mentions free variables is called a free expression. If a free expression is as large as it can be, it is called a maximal free expression, or MFE for short. Note that
sqrt y is actually not technically maximal. Lifting out the MFE would give you:
f x y = let psy = (+) (sqrt y) in g psy x + g psy (2*x) where g psy x = psy x
However, you save no more work here than the second version, and in addition, the resulting function is harder to read. In general, it only makes sense to abstract out a free expression if it is also a reducible expression.